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A355197
Sum of the middle parts of the partitions of n into exactly 3 prime parts.
3
0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 3, 5, 8, 8, 5, 12, 10, 17, 8, 15, 12, 26, 10, 24, 19, 33, 23, 48, 23, 57, 16, 54, 31, 80, 16, 71, 36, 93, 42, 112, 26, 122, 31, 120, 48, 145, 23, 148, 48, 160, 59, 190, 42, 218, 39, 196, 71, 240, 35, 246, 62, 252, 108, 310, 53, 366, 59, 340, 96
OFFSET
0,7
FORMULA
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) * i, where c = A010051.
a(n) = A355199(n) - A355196(n) - A355198(n).
EXAMPLE
a(9) = 5; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of the middle parts of these partitions is 2+3 = 5, we have a(9) = 5.
MATHEMATICA
Table[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1])(PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 23 2022
STATUS
approved